Develop Skills and Strategies Lesson 11 Part 1: ntrouction CCLS ...

Develop Skills and Strategies

Lesson 11

Part 1: Introduction

CCLS

8.EE.B.5

Represent Proportional Relationships

You know how to identify proportional relationships and linear functions. Take a look

at this problem.

Only the top runners in the world can finish a marathon in close to 2 hours. This

graph shows the pace that one such runner followed in a race. What is the unit rate?

Marathon Pace

28

Distance (mi)

24

20

16

12

B

8

A

4

0

0

20

40

60 80 100 120 140

Time (min)

Explore It

Use math you already know to solve the problem.

Write the coordinates for points A and B. 

How do you interpret these points in terms of the number of miles run in a given number

of minutes? 



How can you use the information above to find the rate per minute?



Find the unit rate. Show your work.

What does the unit rate mean in the problem situation? 



100

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Part 1: Introduction

Lesson 11

Find Out More

The unit rate can be represented by the slope of a graph. Compare (0, 0) and (20, 4) in the

graph on the previous page. It shows that the rise is 4 and the run is 20. The slope, or ??rise??

run

¡¤¡¤¡¤

is ?? 4 ?? or ??1???.

20 ¡¤¡¤

5

¡¤¡¤

You can also use two points on the graph to find the slope. Use the formula below with

(0, 0) and (20, 4). Label one of the coordinates (x1, y1) and the other coordinate (x2, y2). Then

substitute the values for the variables in the formula for the slope.

?? y2 2 y1?? 5 ?? 4 2 0 ?? or ?? 4 ?? 5 ?? 1 ??

x2 2 x1

¡¤¡¤¡¤¡¤¡¤¡¤

20 2 0

¡¤¡¤¡¤¡¤¡¤¡¤

20

¡¤¡¤¡¤

5

¡¤¡¤

The y-intercept of the graph on the previous page is 0. The graphs of all proportional

relationships include the point (0, 0), so the y-intercept is always 0.

You have seen that equations of linear functions follow the format y 5 mx 1 b, where m is the

slope and b is the y-intercept. The graphs of proportional relationships also fit this format,

but the value of b is always 0. Let¡¯s look at the equation for the graph on the previous page.

slope

y 5 ??1??? x 1 0

y-intercept

5

¡¤¡¤

5 ?? 1?? x

5

¡¤¡¤

This means that all proportional relationships are linear functions in which the y-intercept is 0.

Reflect

1 Explain how it is possible that all proportional relationships are linear functions but not

all linear functions are proportional relationships.









L11: Represent Proportional Relationships

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101

Part 2: Modeled Instruction

Lesson 11

Read the problem below. Then explore different ways to represent and interpret the

unit rate of a proportional relationship.

A website contains hundreds of different recipes. Included with each recipe is a table

that gives the approximate cost for different numbers of servings. Find the unit cost

for this recipe. Then write an equation for finding the cost of x servings.

Roast Chicken with Vegetables and Rice

Number of Servings

2

4

6

5.00

Cost ($)

2.50

7.50

8

10.00

Picture It

Cost ($)

You can use a graph to represent a proportional relationship.

10

9

8

7

6

5

4

3

2

1

0

Recipe Cost

(1, 1.25)

0

1

2

3 4 5 6 7

Number of Servings

8

9

Model It

You can use an equation to represent the unit rate of a proportional relationship.

The graph shows that 1 serving of the recipe costs $1.25.

You can also divide any of the equivalent ratios in the table: ??2.50

?? 5 $1.25 and ??10?? 5 $1.25.

2

¡¤¡¤¡¤¡¤

8

¡¤¡¤

The equation for the cost of x servings is the unit rate 3 number of servings: y 5 1.25x.

102

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Part 2: Guided Instruction

Lesson 11

Connect It

Now you will use the table, graph, and equation from the previous page to interpret

the unit rate.

2 Describe how the unit rate is shown in the table, graph, and equation.





3 What is the constant of proportionality? How does this relate to the unit rate?



4 What is the slope of the graph? How does this relate to the unit rate and constant of

proportionality?





5 Explain what the unit rate means in the context of the problem.





Try It

6 The table shows the function of how many words Gary can read if he reads at a constant

rate. Use the information in the table to make a graph, representing the number of

minutes on the horizontal axis. Find the slope of the graph and explain what it means in

this situation.

Number of Minutes

Number of Words

2

320

4

6

8

640 960 1,280

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Part 3: Modeled Instruction

Lesson 11

Read the problem below. Then explore different ways to compare two different

proportional relationships.

The table and equation show the rates at which two different printers print in terms of

pages per minute. What is the difference between the per page rate of the two

printers? Which printer is faster?

Printer A

Number of Minutes

1

Number of Pages Printed

40

2

80

3

120

Printer B¡¯s printing

rate is defined by the

equation y 5 50x,

where x is the number

of minutes and y is

the number of pages

printed.

4

160

Picture It

You can use the equation for printer B to make a table of values.

Equation: y 5 50x

Printer B

Number of Minutes

1

Number of Pages Printed

50

2

100

3

150

4

200

Graph It

You can use the tables to graph the proportional relationships.

Printer Rates

Number of Pages Printed

160

140

120

Printer B

80

60

40

20

0

104

Printer A

100

0

1

2

3

4

Number of Minutes

5

L11: Represent Proportional Relationships

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